Implications

This non-implication, Form 167 \( \not \Rightarrow \) Form 344, whose code is 4, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1091, whose string of implications is:
    71-alpha \(\Rightarrow\) 9 \(\Rightarrow\) 376 \(\Rightarrow\) 167
  • A proven non-implication whose code is 3. In this case, it's Code 3: 1366, Form 71-alpha \( \not \Rightarrow \) Form 344 whose summary information is:
    Hypothesis Statement
    Form 71-alpha  <p> \(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). <a href="/books/8">Jech [1973b]</a>, page 119. </p>

    Conclusion Statement
    Form 344 <p> If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\). </p>

  • This non-implication was constructed without the use of this last code 2/1 implication

The conclusion Form 167 \( \not \Rightarrow \) Form 344 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N2(\aleph_{\alpha})\) Jech's Model This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\)

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